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In geometry, an affine transformation, affine map or an affinity (from the Latin, affinis, "connected with") is a …

An image of a fern-like fractal that exhibits affine self-similarity. Each of the leaves of the fern is related to each other leaf by an affine transformation. For instance, the red leaf can be transformed into both the small dark blue leaf and the large light blue leaf by a combination of reflection, rotation, scaling, and translation.

Image: Affine Transformation Original Checkerboard

Image: Affine Transformation Reflected Checkerboard

Image: Affine Transformation Scale Checkerboard

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1. Affine transformation – In geometry, an affine transformation, affine map or an affinity is a function between affine spaces which preserves points, straight lines and planes. Also, sets of parallel lines remain parallel after an affine transformation, an affine transformation does not necessarily preserve angles between lines or distances between points, though it does preserve ratios of distances between points lying on a straight line. Examples of affine transformations include translation, scaling, homothety, similarity transformation, reflection, rotation, shear mapping, and compositions of them in any combination and sequence. If X and Y are affine spaces, then every affine transformation f, X → Y is of the form x ↦ M x + b, unlike a purely linear transformation, an affine map need not preserve the zero point in a linear space. Thus, every linear transformation is affine, but not every affine transformation is linear, all Euclidean spaces are affine, but there are affine spaces that are non-Euclidean. In affine coordinates, which include Cartesian coordinates in Euclidean spaces, another way to deal with affine transformations systematically is to select a point as the origin, then, any affine transformation is equivalent to a linear transformation followed by a translation. An affine map f, A → B between two spaces is a map on the points that acts linearly on the vectors. In symbols, f determines a linear transformation φ such that and we can interpret this definition in a few other ways, as follows. If an origin O ∈ A is chosen, and B denotes its image f ∈ B, the conclusion is that, intuitively, f consists of a translation and a linear map. In other words, f preserves barycenters, as shown above, an affine map is the composition of two functions, a translation and a linear map. Ordinary vector algebra uses matrix multiplication to represent linear maps, using an augmented matrix and an augmented vector, it is possible to represent both the translation and the linear map using a single matrix multiplication. If A is a matrix, = is equivalent to the following y → = A x → + b →, the above-mentioned augmented matrix is called an affine transformation matrix, or projective transformation matrix. This representation exhibits the set of all affine transformations as the semidirect product of K n and G L. This is a group under the operation of composition of functions, ordinary matrix-vector multiplication always maps the origin to the origin, and could therefore never represent a translation, in which the origin must necessarily be mapped to some other point. By appending the additional coordinate 1 to every vector, one considers the space to be mapped as a subset of a space with an additional dimension. In that space, the original space occupies the subset in which the coordinate is 1. Thus the origin of the space can be found at. A translation within the space by means of a linear transformation of the higher-dimensional space is then possible

2. Fractal – A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known as expanding symmetry or evolving symmetry, if the replication is exactly the same at every scale, it is called a self-similar pattern. An example of this is the Menger Sponge, Fractals can also be nearly the same at different levels. This latter pattern is illustrated in small magnifications of the Mandelbrot set, Fractals also include the idea of a detailed pattern that repeats itself. Fractals are different from other geometric figures because of the way in which they scale, doubling the edge lengths of a polygon multiplies its area by four, which is two raised to the power of two. Likewise, if the radius of a sphere is doubled, its volume scales by eight, but if a fractals one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer. This power is called the dimension of the fractal. As mathematical equations, fractals are usually nowhere differentiable, the term fractal was first used by mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus meaning broken or fractured, there is some disagreement amongst authorities about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as beautiful, damn hard, increasingly useful, Fractals are not limited to geometric patterns, but can also describe processes in time. Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds and found in nature, technology, art, Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractal. The word fractal often has different connotations for laypeople than for mathematicians, the mathematical concept is difficult to define formally even for mathematicians, but key features can be understood with little mathematical background. If this is done on fractals, however, no new detail appears, nothing changes, self-similarity itself is not necessarily counter-intuitive. The difference for fractals is that the pattern reproduced must be detailed, a regular line, for instance, is conventionally understood to be 1-dimensional, if such a curve is divided into pieces each 1/3 the length of the original, there are always 3 equal pieces. In contrast, consider the Koch snowflake and it is also 1-dimensional for the same reason as the ordinary line, but it has, in addition, a fractal dimension greater than 1 because of how its detail can be measured. This also leads to understanding a third feature, that fractals as mathematical equations are nowhere differentiable, in a concrete sense, this means fractals cannot be measured in traditional ways. The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, according to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity. In his writings, Leibniz used the term fractional exponents, also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called self-inverse fractals

3. Geometry – Geometry is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer, Geometry arose independently in a number of early cultures as a practical way for dealing with lengths, areas, and volumes. Geometry began to see elements of mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into a form by Euclid, whose treatment, Euclids Elements. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC, islamic scientists preserved Greek ideas and expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid footing by mathematicians such as René Descartes. Since then, and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, while geometry has evolved significantly throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, lines, planes, surfaces, angles, contemporary geometry has many subfields, Euclidean geometry is geometry in its classical sense. The mandatory educational curriculum of the majority of nations includes the study of points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, Euclidean geometry also has applications in computer science, crystallography, and various branches of modern mathematics. Differential geometry uses techniques of calculus and linear algebra to problems in geometry. It has applications in physics, including in general relativity, topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this often means dealing with large-scale properties of spaces, convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis. It has close connections to convex analysis, optimization and functional analysis, algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques. It has applications in areas, including cryptography and string theory. Discrete geometry is concerned mainly with questions of relative position of simple objects, such as points. It shares many methods and principles with combinatorics, Geometry has applications to many fields, including art, architecture, physics, as well as to other branches of mathematics. The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, the earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, later clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiters position and motion within time-velocity space

4. Computer graphics – Computer graphics are pictures and films created using computers. Usually, the term refers to computer-generated image data created with help from specialized hardware and software. It is a vast and recent area in computer science, the phrase was coined in 1960, by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, though sometimes referred to as CGI. The overall methodology depends heavily on the sciences of geometry, optics. Computer graphics is responsible for displaying art and image data effectively and meaningfully to the user and it is also used for processing image data received from the physical world. Computer graphic development has had a significant impact on many types of media and has revolutionized animation, movies, advertising, video games, the term computer graphics has been used a broad sense to describe almost everything on computers that is not text or sound. Such imagery is found in and on television, newspapers, weather reports, a well-constructed graph can present complex statistics in a form that is easier to understand and interpret. In the media such graphs are used to illustrate papers, reports, thesis, many tools have been developed to visualize data. Computer generated imagery can be categorized into different types, two dimensional, three dimensional, and animated graphics. As technology has improved, 3D computer graphics have become more common, Computer graphics has emerged as a sub-field of computer science which studies methods for digitally synthesizing and manipulating visual content. Screens could display art since the Lumiere brothers use of mattes to create effects for the earliest films dating from 1895. New kinds of displays were needed to process the wealth of information resulting from such projects, early projects like the Whirlwind and SAGE Projects introduced the CRT as a viable display and interaction interface and introduced the light pen as an input device. Douglas T. Ross of the Whirlwind SAGE system performed an experiment in 1954 in which a small program he wrote captured the movement of his finger. Electronics pioneer Hewlett-Packard went public in 1957 after incorporating the decade prior, and established ties with Stanford University through its founders. This began the transformation of the southern San Francisco Bay Area into the worlds leading computer technology hub - now known as Silicon Valley. The field of computer graphics developed with the emergence of computer graphics hardware, further advances in computing led to greater advancements in interactive computer graphics. In 1959, the TX-2 computer was developed at MITs Lincoln Laboratory, the TX-2 integrated a number of new man-machine interfaces